Elsevier

Automatica

Volume 69, July 2016, Pages 342-347
Automatica

Brief paper
Combined economic and regulatory predictive control

https://doi.org/10.1016/j.automatica.2015.12.003Get rights and content

Abstract

A combined economic and regulatory Model Predictive Control (MPC) framework, referred to as dual-objective MPC, is presented. Economic and regulatory cost objectives are combined using an explicitly defined dynamic weight function. The features embedded in the weight function allow for increased economic online performance during process transients, while simultaneously steering the process to an optimal economic steady-state. The proposed strategy gives an online optimization problem of same type as typical nonlinear MPC formulations, with the same number of variables. Increased economic transient performance is illustrated for an acetylene hydrogenation case, when compared to regulatory MPC. A link to dissipativity theory for the presented dual-objective MPC strategy concludes the work.

Introduction

Economic Model Predictive Control (economic MPC) has emerged as a possible core component of the next-generation process control framework. Economic MPC reconciles real-time economic process optimization with process control, solving optimization problems with general economic cost functions. Observed economic benefits of economic MPC over regulatory MPC are now well documented (Amrit, 2011, Rawlings et al., 2012); however, solutions to challenges such as stability, guaranteed closed-loop economic performance and computational considerations are still at an early stage (Angeli et al., 2012, Diehl et al., 2011, Müller et al., 2013). For general economic cost functions, economic MPC retains the property of average asymptotic performance being at least as good as the best admissible steady-state (Angeli, Amrit, & Rawlings, 2009); however, stability is not necessarily guaranteed (Rawlings et al., 2012). To establish asymptotic stability in the context of economic MPC, one requires the assumption of strict dissipativity to hold (Angeli et al., 2012). It has been observed, through simulation, that convergence of economic MPC is attainable if one extends the economic cost function with a sufficiently convex term (Angeli, Amrit, & Rawlings, 2011). In the context of dissipativity theory, the addition of a sufficiently convex term transforms the system to be strictly dissipative and asymptotic stability of the origin can be concluded (Angeli et al., 2012). The extension of an economic cost function with some convex cost term in the work of Angeli et al. (2011), Angeli et al. (2012) draws close parallels to combined economic and regulatory MPC (Maree & Imsland, 2011), also referred to as dual-objective MPC in the present work. In Maree and Imsland (2011), a weight is incorporated to combine the respective objectives in a convex fashion. Specifying, however, the exact nature of such a weight, such that stability and good closed-loop performance are ensured, is not immediately clear (Rawlings et al., 2012).

Maree and Imsland (2014) proposed sufficient conditions for a dynamic weight function, which if satisfied in the context of dual-objective MPC, promote increased economic performance during transients, and simultaneously ensure that the origin is asymptotically stable under the receding horizon control law. The present work relaxes the conditions required for stability in the aforementioned work. Instead, less-restrictive conditions are proposed, which if satisfied, admits convergence of closed-loop trajectories towards some admissible steady-state. The latter property, in the context of economic MPC, can be interpreted as a safe-park property (Mhaskar, Liu, & Christofides, 2013), and may be desirable during uncertain process conditions or severe disturbances that necessitates conservative process regulation in the vicinity of some admissible steady-state. Additionally, we give a theoretical link between the conditions on the dynamic weight function and the work of Angeli et al. (2012) on the role of dissipativity in economic MPC.

Theoretical concepts are illustrated for an acetylene hydrogenation case. The strength of the proposed dual-objective MPC strategy, with dynamic weighting, is simplicity of numerical implementation, with computational complexity comparable with online optimization problems of same type as nonlinear MPC formulations, with the same number of variables.

Notation

The symbols I and R define integer and real numbers, respectively. A continuous function γ:R0R0 belongs to a class-K function if γ(0)=0 and is strictly increasing. A continuous function φ:RR0 belongs to a class-PD (is positive definite) if φ(s)>0,s0 and φ(0)=0. A continuous function β:I0×R0R0 belongs to a class-PDL function if β(t;)PD, and β(t;s) is nonincreasing in t, satisfying limtβ(t;s)=0. The Euclidean norm for a vector argument is defined as ||.

Section snippets

Preliminaries

We consider the nonlinear, discrete-time system model x+=f(x,u) with state, xXRnx, and control input uURnu. We assume that (1) is continuous in (x,u). Define the mixed constraint set (x(k),u(k))Z,kI0 for a compact set Z=X×U. Let le:ZR be a continuous economic stage cost function. Then, the optimal solution to the steady-state optimization problem (xs,us)argmin(x,u)Z{le(x,u)|x=f(x,u)} defines the economic steady-state, (xs,us). The existence of (xs,us) follows from the continuity of

Combined regulatory and economic predictive control

Consider, in this section, any general continuous weight function, μ:Z[0,1]. A specific weight function, fulfilling the assumptions made here and in Section  4, will be proposed in Section  5. The combined regulatory and economic stage cost function (henceforth referred to as the dual-objective stage cost function) is defined lμ(x,u)μ(x,u)le(x,u)+[1μ(x,u)]lr(x,u). We also define the terminal set Xf being a subset of X containing a neighbourhood of the economic optimal steady-state, xs. In

Stability and convergence analysis

In this section we concern ourselves with analysing the stability and convergence of (7) with a terminal state constraint region being defined XfX, in which Vf(x)0,xXf. The special case Xf{xs}, in which Vf(x)=0,xXf, is omitted for brevity; however, the reader is referred to Maree (2014) for a rigorous analysis. Two assumptions to the former definition of Xf (in the context ofAssumption 1 are proposed, which if satisfied, enable us to either conclude on asymptotic stability of, or the

Economic optimal weight functions

Section  4 stipulated necessary assumptions for a dual-objective stage cost function lμ(), which if satisfied, guarantees the economic optimal steady-state point xs to be asymptotically stable for the closed-loop trajectories (8) with region of attraction X0. Under relaxed conditions, one is still able to show convergence of trajectories (8) to xs. In the context of optimizing for economic performance, it is desirable to choose μ() in (4) as close to one as possible. We present a strategy in

Numerical example

Acetylene hydrogenation possesses stoichiometry which may be characterized as consecutive–competitive process reactions. If these reactions are conducted in an isothermal catalytic continuous stirred tank reactor, then the dynamic behaviour of the process can be described by following dimensionless form (Lee & Bailey, 1980): ẋ1=1x1a;ẋ2=ux2ab;ẋ3=x3+abaσ1x1x21+βx1;bσ2x20.5x31+βx1. We define x1 and x2 as the concentration of acetylene and hydrogen in the feed, respectively. State x3 is

A link to dissipativity

Asymptotic stability of xs, in the context of economic MPC, can be shown if the nonlinear system (1) is strictly dissipative with respect to an economic supply rate, le(x,u) (Angeli et al., 2012). If this holds, then we can show, in the context of dual-objective MPC, that (1) will also be strictly dissipative with respect to the dual-objective supply rate, lμ(x,u), given μ() is chosen according to Proposition 1. Such a result allows, under the dissipativity assumption, to conclude on

Concluding remarks

The present work proposed a combined dual-objective MPC framework in which economic and regulatory objectives are combined in a convex fashion making use of a dynamic weight function. Special conditions, explicitly embedded in the dynamic weight function, ensure that benefits of both economic MPC and regulatory MPC are observed during the online implementation of the resulting dual-objective receding horizon control law. Economic performance is optimized during process transients, with

Johannes Philippus Maree received Ph.D. degree in Engineering Cybernetics from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2014. As a part of his Ph.D. studies, he was a visiting scholar at the Department of Chemical and Biological Engineering, University of Wisconsin-Madison, USA. His research interests include nonlinear control, estimation and system analysis, numerical optimization and embedded model predictive

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Johannes Philippus Maree received Ph.D. degree in Engineering Cybernetics from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2014. As a part of his Ph.D. studies, he was a visiting scholar at the Department of Chemical and Biological Engineering, University of Wisconsin-Madison, USA. His research interests include nonlinear control, estimation and system analysis, numerical optimization and embedded model predictive control.

Lars Imsland received Ph.D. degree in Electrical Engineering from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2002. As a part of his Ph.D. studies, he was a visiting scholar at the Institute for Systems Theory in Engineering at the University of Stuttgart, Germany. After his Ph.D. he worked as a postdoctoral researcher at NTNU, a research scientist at SINTEF and as a specialist for Cybernetica AS, before becoming a Professor in Control Engineering at NTNU in 2009. His main research interests are the theory and application of nonlinear and optimizing control and estimation.

The material in this paper was partially presented at the 19th IFAC World Congress, August 24–29, 2104, Cape Town, South Africa. This paper was recommended for publication in revised form by Associate Editor David Angeli under the direction of Editor Andrew R. Teel.

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